A subsequence principle characterizing Banach spaces containing 𝑐₀

Rosenthal, Haskell

doi:10.1090/S0273-0979-1994-00494-X

A subsequence principle characterizing Banach spaces containing $c_0$
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Bull. Amer. Math. Soc. 30 (1994), 227-233 Request permission

Abstract:

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent to the summing basis. (A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.) The following characterization of spaces containing ${c_0}$ is thus obtained, in the spirit of the author’s 1974 subsequence principle characterizing Banach spaces containing ${\ell ^1}$. Corollary 1. A Banach space B contains no isomorph of ${c_0}$ if and only if every non-trivial weak-Cauchy sequence in B has a strongly summing subsequence. Combining the ${c_0}$-and ${\ell ^1}$-theorems, one obtains Corollary 2. If B is a non-reflexive Banach space such that ${X^{\ast }}$ is weakly sequentially complete for all linear subspaces X of B, then ${c_0}$ embeds in B.

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A characterization of Banach spaces containing ${c_0}$

Differences of bounded semi-continuous functions